Problem 33 - Algebra 2

Question

Some say that to brew an excellent cup of coffee, you must have a brewing temperature of \(200^{\circ} \mathrm{F}\) , plus or minus 5 degrees. Write and solve an equation describing the maximum and minimum brewing temperatures for an excellent cup of coffee.

Step-by-Step Solution

Answer

The brewing temperature must be between 195°F and 205°F.

Step 1: Understanding the Temperature Range

The problem states that the ideal brewing temperature for coffee is 200°F, with a possible variation of ±5°F. This means the temperature range for brewing coffee can be expressed mathematically.

Step 2: Setting up the Equation

To express this range, we can create the inequality: \[200 - 5 \leq T \leq 200 + 5\]where \(T\) represents the actual brewing temperature.

Step 3: Solving for Minimum Temperature

Solve the inequality for the minimum side by subtracting: \[T \geq 200 - 5\]which simplifies to \(T \geq 195\). Therefore, the minimum temperature is 195°F.

Step 4: Solving for Maximum Temperature

Solve the inequality for the maximum side by adding: \[T \leq 200 + 5\]which simplifies to \(T \leq 205\). Therefore, the maximum temperature is 205°F.

Step 5: Conclusion

The brewing temperature for an excellent cup of coffee must be within the range of 195°F to 205°F.

Key Concepts

Temperature Range

When brewing coffee, the temperature range plays a crucial role in determining the quality of the drink. The term "temperature range" refers to the spectrum of temperatures at which a specific process, like coffee brewing, can be optimally performed. In this context, the problem provides an ideal temperature of 200°F with a variation of ±5°F. This variation means that the temperature can be 5 degrees above or below the ideal value, giving us a range for brewing the coffee. Understanding a temperature range involves recognizing the limits within which a certain activity can be performed successfully. In our exercise, this means we must ensure the actual brewing temperature is between 195°F and 205°F. Here is what that looks like step-by-step:

• The base temperature provided is 200°F.
• The allowable deviation is ±5°F, leading us to calculate both a lower and upper threshold.
• The lower threshold is 200°F - 5°F, resulting in 195°F.
• The upper threshold is 200°F + 5°F, resulting in 205°F.

Each of these calculations helps us understand and define the temperature range essential for an excellent cup of coffee. This concept is widely applicable in various scientific and culinary contexts where precise control of conditions is desired.

Inequality Solving

Inequality solving is a mathematical technique used to find the range of values that satisfy a given inequality. In our coffee brewing exercise, we use an inequality to express and solve for the permissible temperature range.The initial inequality for temperature can be written as:\[200 - 5 \leq T \leq 200 + 5\]We solve this inequality by addressing each side separately:

• Minimum Temperature:
  The inequality on the left side is solved by subtracting 5 from 200, which represents the smallest permissible temperature (T ≥ 195°F).
• Maximum Temperature:
  The inequality on the right side requires adding 5 to 200, indicating the maximum permissible temperature (T ≤ 205°F).

By solving these inequalities, we determine that the brewing temperature must lie between 195°F and 205°F for optimal coffee brewing. This approach is commonly used whenever boundaries or limits need to be defined through inequalities in mathematics.

Mathematical Expressions

Mathematical expressions are central to formulating problems and expressing solutions in a way that is both clear and precise. In the case of the coffee brewing exercise, the expression \[200 - 5 \leq T \leq 200 + 5\] is used to define the acceptable temperature range. Understanding mathematical expressions involves recognizing how numbers, variables, and operations are combined to express relationships or constraints. Here are the key components involved:

• Numerical values:
  The numbers 200, 5, 195, and 205 are used explicitly to set the parameters of the brewing temperature.
• Variable:
  "T" serves as the variable representing the actual brewing temperature.
• Operators:
  The operators ≤ and ≥ show us that the value must fall within a specific boundary or limit.

By comprehending these components, one can easily manipulate or interpret expressions, making it an invaluable skill in various areas of math and real-world applications such as this coffee brewing example. This skill enhances our ability to model situations mathematically and derive meaningful solutions.